University of Colorado Boulder

Math 5001, Midterm, Part 2, Take-Home

Fall 2019

NAME:

Question Points Score

1 20

2 20

Total: 40

• Only the course textbook by Rudin, lecture notes and your HW solutions are allowed.

• Read instructions carefully. Show all your reasoning and work for full credit unless indicated otherwise.

• On my honor, as a University of Colorado Boulder student, I have neither given nor received unautho-rized assistance.

• You can type or handwrite your answers. Copy the statement of each problem.

& Math 4001

solutions for both.

grad sign!;? -

d

Math 5001 Midterm, Part 2, Take-Home Fall 2019

1. (20 points) IfP

n an andP

n bn are two series of nonnegative real numbers, prove thatP

n an andP

n bn converge if and only ifP

n a2n + b2n converges.

2. (20 points) (a) (15pts) Prove or disprove: If {fn} is a sequence of continuous functions

converging uniformly on [0, 1] to f , then

limn!1

Z 1� 1n

0

fn(x)dx =

Z 1

0

f(x)dx.

(b) (5pts) Prove or disprove: If each fn � 0 on [0, 1] and {fn} is a sequence of continuousfunctions such that

Pn fn converge uniformly on [0, 1] to f , then

1X

n=1

Z 1� 1n

0

fn(x)dx =

Z 1

0

f(x)dx.

Page 1 of 1

-

I. By HW Problem 1.10

,we have 7 constants

c,

d > O such that if x = ra ,b ) then

CHXH,SH Ma Sdllxll,

⇐c ( at b) STATE b drat b ) CA)

( Can also reprovethis directly now : for this problem we need

Osa ,b 's TEs at b 88

By given o sa ,b

.

Also a= la I = ta s TE since 650

.

Also EE s at b ⇐ > aft b- Sfat b )

2

x' & Tx are increasing

fu wetters for x 30.

Now suppose § an& { bn converge .

We could use Th -3.24

,but we will just use

the

comparisontest .

Fit E ant bn conveys( do a

direct pout o- g

notte Thy .gg )

Then I Ebn I = Esh s an tbh H n by @ * ).

⇒ The series Eam converges by the conparbefnesf.

Now suppose ETain converges .

By God Ian Is anstab

.

SO Ean conveys by the comparisontest

.

Similarly ,the sane holds for [ bn since

I but Sarin.

2 a)

Proof : The statement is false .

Consider fnrx ) = InXcagthen Efnrx)converges

uniformly to fro ' Xco, ,y by he geometric series

.

We then have b I I

E f In

dxsfotadxi-%afi-ndx.fi

I do =L by HUI

us ,d Corollary p

.

152 .

Now consider

I,

so' 't

frodo .So

' "

'sdxdxtn.at"

's do

= E. so' "

Into ¥1

b/c fo'

lady s 's & So

' "

's do ' O